In the exercise, W is a standard, one-dimensional Brownian motion and $0 \lt T \lt \infty$. We are asked to show that $$\lim_{\beta\rightarrow\infty}\sup_{0\le t\le T}|e^{-\beta t}\int_0^t e^{\beta s}dW_s|=0,\ \ a.s.$$ I know that $E|e^{-\beta t}\int_0^t e^{\beta s}dW_s|^2$ can be calculated, and to deal with "sup", I hope to use BDG inequality or Doob's inequality. However, they can only be applied to (sub or super)martingales. So, I try to do some transformation to the integral. Time-change can make it a Brownian motion as $B_{{1\over2\beta}(e^{2t}-1)}$, and local time gives its decomposition as the sum of a martingale and a increasing process. For the martingale part, I could use Doob's inequality. For the other part, I guess that the increasing process cannot increase as fast as exponential function. I think it may be a way to solve the problem, but I cannot get the estimate about local time.
If you have some idea about this exercise itself, or my problem about the estimation of the local time, please share with me. Thank you!